Properties

Label 2-378-63.59-c1-0-6
Degree $2$
Conductor $378$
Sign $-0.589 + 0.807i$
Analytic cond. $3.01834$
Root an. cond. $1.73733$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + 0.0676·5-s + (−2.64 − 0.142i)7-s − 0.999i·8-s + (−0.0585 − 0.0338i)10-s − 3.92i·11-s + (−3.32 − 1.92i)13-s + (2.21 + 1.44i)14-s + (−0.5 + 0.866i)16-s + (0.775 − 1.34i)17-s + (5.06 − 2.92i)19-s + (0.0338 + 0.0585i)20-s + (−1.96 + 3.40i)22-s − 5.52i·23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + 0.0302·5-s + (−0.998 − 0.0538i)7-s − 0.353i·8-s + (−0.0185 − 0.0106i)10-s − 1.18i·11-s + (−0.922 − 0.532i)13-s + (0.592 + 0.385i)14-s + (−0.125 + 0.216i)16-s + (0.188 − 0.325i)17-s + (1.16 − 0.670i)19-s + (0.00755 + 0.0130i)20-s + (−0.418 + 0.725i)22-s − 1.15i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.589 + 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.589 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.589 + 0.807i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ -0.589 + 0.807i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.278693 - 0.548746i\)
\(L(\frac12)\) \(\approx\) \(0.278693 - 0.548746i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 \)
7 \( 1 + (2.64 + 0.142i)T \)
good5 \( 1 - 0.0676T + 5T^{2} \)
11 \( 1 + 3.92iT - 11T^{2} \)
13 \( 1 + (3.32 + 1.92i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.775 + 1.34i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.06 + 2.92i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 5.52iT - 23T^{2} \)
29 \( 1 + (1.20 - 0.697i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.09 + 0.632i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.35 + 7.54i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.17 - 8.96i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.735 - 1.27i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.77 + 3.06i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.28 + 3.63i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.70 + 8.14i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.0705 - 0.0407i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.67 - 13.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.30iT - 71T^{2} \)
73 \( 1 + (-6.12 - 3.53i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.42 + 5.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.93 - 6.81i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.84 - 10.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.363 + 0.209i)T + (48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.97280450001121776740728085350, −9.979527733762324193494483945126, −9.411033380424250780930792391706, −8.365729034998887177102329442465, −7.38913293900324709781447671004, −6.39315277843936569533975673616, −5.23454621832684618329625751951, −3.55498623988626173895428104043, −2.64819859890814357481686464107, −0.48517333819357693020467293305, 1.89530818684079336477521485101, 3.50142964724486290321779052247, 5.00249974643381098908249282500, 6.11110581066634066740825398551, 7.16616750284267041008135881298, 7.74198330599698336147031774030, 9.216504578748515286160187187427, 9.724164739921102162450157437485, 10.36742921575094276303240853556, 11.86969387948996097569923629933

Graph of the $Z$-function along the critical line